1. Field of the Invention
This invention pertains to Fabry-Perot etalons. More particularly, this invention relates to holographic Fabry-Perot etalons and a method for making high finesse Fabry-Perot etalons.
2. Background of the Prior Art
Fabry-Perot etalons based on two mirror systems separated by a spacer have been extensively discussed in the literature. See M. Born and E. Wolf, "Principles of Optics" Pergamon Press (1980) incorporated herein by reference. The mirrors used can be of different types--metallic, dielectric multi-layer, or other mirroring surfaces. The spacers used can also be of varying types including air, ordinary plastics, glass plates, and electrooptic (E-O) materials such as liquid crystal (LC), PLZT (PbLa(TiZr)O.sub.3 and piezoelectric material, and others. The general theory of conventional Fabry-Perot etalons is very well known and can be based on wave theory or an optical multi-reflection model. The crucial parameters of Fabry-Perot etalon theory are the individual mirror reflectivities, R, and optical path length defining multi-reflection phase relation of optical beams reflected numerous times between the two mirrors. This can be described by the following equation: ##EQU1## where .lambda..sub.o is the wavelength of light in vacuum, .theta. is the angle of incidence, n is the refractive index of the spacer, and d is spacer thickness. In the absence of optical losses, the Fabry-Perot etalon intensity reflection coefficient R.sub.FP has the form: ##EQU2## where R is the intensity reflection coefficient of a single mirror, assuming both mirrors are identical. Analogously, the Fabry-Perot etalon intensity transmission coefficient has the following form: ##EQU3## From Equations 2 and 3, it is seen that due to the phase coupling of the two mirrors, such that .delta.=n.pi., where n is any even integer, the etalon reflectivity is zero and the etalon transmission is 1.0, even though the individual mirrors may be highly reflective. Following Equation 3, etalon transmission versus optical frequency EQU .nu.=C.sub.o /.lambda..sub.o,
where c is the speed of light in vacuum, has the following approximate form as shown in FIG. 1. In FIG. 1, .DELTA..nu. is the free spectral range (FSR) and .delta..nu. is the full width at half maximum (FWHM). The finesse, F, of the etalon is defined as the ratio of ##EQU4## Assuming that all etalon interfaces are perfectly flat, the finesse parameter has the following form: ##EQU5## Analogously, the contrast factor C is the ratio of the maximum to minimum transmittance of the Fabry-Perot etalon and is as follows:, ##EQU6## Assuming high mirror reflectances of the Fabry-Perot etalon (R&gt;0.9), the above two formulas can be simplified to the following useful forms: ##EQU7## Equation 8 can be used to determine the minimum transmittance and the maximum optical density which characterize the maximum rejection of the optical beam by the etalon. Now we obtain ##EQU8##
The filtering properties of the etalon are determined by the etalon linewidths in the form: ##EQU9## wherein the linewidth .delta..nu. is related to the FWHM as follows: ##EQU10## Using Equations 10 and 11 the thickness l required in order to obtain predetermined line widths can be determined as follows: ##EQU11##
For example, in order to obtain .delta..lambda.=2.ANG. and .DELTA..lambda.=13.9 nm, the required finesse F is ##EQU12## and for .lambda.=0.5.mu. and n=1.5 the required thickness l of the spacer is 6.mu.m. The relationship between etalon linewidths and angular field of view (FOV) can be obtained from Equation 1 in the form: ##EQU13## This relation is reflected in Table 1:
TABLE 1 ______________________________________ .delta..lambda. 10 nm 1 nm 5.ANG. 2.ANG. 1.ANG. ______________________________________ .theta. .14 .04 .03 .02 .01 ______________________________________
The following important relationship defines the influence of spacer flatness on etalon finesse F. ##EQU14## where .sigma..sub.l is root mean square (RMS) of spacer flatness. If Equation 14 is satisfied, then the fundamental parameter defining etalon finesse is the reflection coefficient R, and Equation 5 holds. Otherwise, the fundamental parameter determining etalon finesse is surface flatness determined by RMS .sigma..sub.l. It is useful to replace the RMS parameter by ##EQU15## where N defines the fraction of a wavelength characterizing surface flatness. For example, for .lambda.=1.mu. and N=10, the surface flatness RMS is 0.1.mu. which is a typical flatness for good optical quality surfaces. Using Equation 15, Equation 14 can be transformed into: ##EQU16## This equation is illustrated in Table 2.
TABLE 2 ______________________________________ R .8 .85 .9 .95 .99 F 14 19 30 61 312 N 93 126 189 378 1890 ______________________________________
It is seen from this table that even obtaining moderate finesse etalons requires interfaces having very high surface flatness, greatly exceeding typical high optical quality surfaces. For example, in order to obtain a moderate finesse etalon of F=61, N must be 378. That is, the surface roughness must only be 1/378 of the wavelength .lambda., which for .lambda.=500 nm is on the order of a nanometer. This extremely high smoothness obviously cannot be obtained by known low-cost polishing techniques. It would be highly beneficial to have a method of manufacturing high finesse etalons using only moderately smooth interfaces on the order of N =10.